Quantum mechanics is just deBroglie's hypothesis plus series expansion?

Question

It turns out that it corresponds well with experimental data to say that particles behave like interfering complex distributions, whose modulus squared is a probability mass function for detecting the particle.

The collection of solutions to the Schrodinger equation for a particle with some mass and unspecified energy in a thin infinite-barrier well are of the form: $f_n(x)=\sin(n \pi x/L)$, or any convergent weighted sum of $f_n$s.

However, any 'nice enough' function $f:[0,L]\to\mathbb{C}$ can be well-approximated by such sums. So by superimposing enough energy states, wavefunctions can look like basically anything. So where is the 'quanta'?

It seems to me that in this example, we are using Fourier decomposition to appropriate a discrete aspect onto the analysis of wavefunctions. It is not clear that 'discrete energy states' are intrinsic to nature, not just pieces of the wavefunction's series expansion. What gives?

Answer 1

The collection of solutions to the Schroedinger equation for a particle with some mass and unspecified energy in a thin infinite-barrier well are of the form: $f_n(x)=\sin(n\pi x/L)$, or any convergent weighted sum of fns.

Under some suitable conditions any function can be approximated as expansion over that basis. What distinguishes different solutions of different equations (and therefore different dynamics) are the boundary conditions and, most important, the coefficients of that expansions (notice that an element in a vector space is uniquely identified by the set of coefficients it possesses over a certain basis - the same holds mutatis mutandis in more complicated space structures).

In the particular case at hand the coefficients of the expansion can be usually determined by solving the Schroedinger equation. In the most general case the Fourier transform of the wave function and the wave function itself have a certain relation that contains some physical information related to the representations onto the position and momentum eigenstates.

It is not clear that 'discrete energy states' are intrinsic to nature, not just pieces of the wavefunction's series expansion. What gives?

Energy eigenstates are eigenstates of the Hamiltonian operator and their form is uniquely (if some conditions hold) determined by the operator itself and its domain. There is an entire area of mathematics that deals with spectral theory of self-adjoint operators and what the eigenstates must look like.

So where is the 'quanta'?

There is no "quanta" is non-relativist Quantum Mechanics, except if you want to define them as the discrete energy spectrum. Quanta naturally emerge in Quantum Field Theory instead, as force carriers of the gauge fields, but that is another matter.